Issue with math in physics problem?

[fightboy]

Ok so for the problem I'm mainly having trouble with setting up one of the equations.
The solutions manual jumped from:
ω_iz([(2/5m_ER_E²)/(2/5m_ER_E²) + (2/3m_debrisR_E²)]-1)
to:
ω_iz([m_E/(m_E + (5/3m_debris))]-1)

I placed the brackets to clearly separate the fraction from the -1. Anyways I'm having trouble seeing how the math is done to get from the first equation to the second equation, how was it simplified? If anyone is confused on how i wrote the problem please just ask! Thanks!

 

[Mark44]

Ok so for the problem I'm mainly having trouble with setting up one of the equations.
The solutions manual jumped from:
ω_iz([(2/5m_ER_E²)/(2/5m_ER_E²) + (2/3m_debrisR_E²)]-1)
to:
ω_iz([m_E/(m_E + (5/3m_debris))]-1)

I placed the brackets to clearly separate the fraction from the -1. Anyways I'm having trouble seeing how the math is done to get from the first equation to the second equation

These aren't equations - they are expressions.

Inside the parentheses you have (2/5m_ER_E²)/(2/5m_ER_E²), which is just 1.

The fractions you wrote are ambiguous, which doesn't change what I wrote above.
Is the numerator $\frac{2}{5}m_ER_E^2$
or is it $\frac{2}{5m_ER_E^2}$?

, how was it simplified? If anyone is confused on how i wrote the problem please just ask! Thanks!

 

[fightboy]

These aren't equations - they are expressions.

Inside the parentheses you have (2/5m_ER_E²)/(2/5m_ER_E²), which is just 1.

The fractions you wrote are ambiguous, which doesn't change what I wrote above.
Is the numerator $\frac{2}{5}m_ER_E^2$
or is it $\frac{2}{5m_ER_E^2}$?

The numerator is $\frac{2}{5}m_ER_E^2$. I get that $\frac{2}{5}m_ER_E^2$/$\frac{2}{5}m_ER_E^2$ is equal to 1, I am mainly confused on how (2/3m_debrisR_E²) became (5/3m_debris) or if the manual made a mistake.

 

[gopher_p]

If you put an extra set of parentheses in the original expression, the $R_E^2$s cancel and you can multiply everything by $\frac{5}{2}$ to get $$\frac{\frac{2}{5}m_ER_E^2}{\frac{2}{5}m_ER_E^2+\frac{2}{3}m_{\text{debris}} R_E^2}=\frac{m_E}{m_E+\frac{5}{3}m_{\text{debris}}}$$

 

[CellCoree]

I understand that math can be challenging and it's important to have a clear understanding of the equations we use in physics. In this case, it seems like there may have been a step or two missing in the solutions manual. It would be helpful to see the entire problem and the steps leading up to the first equation in order to fully understand the simplification process.

One possible explanation for the simplification could be that the term (2/5mERE2) was factored out and then the remaining terms were combined to create the second equation. However, without more context it is difficult to say for sure.

It's always important to double check your work and make sure you understand each step in a problem. If you are still having trouble, I would suggest reaching out to your teacher or a tutor for clarification. Additionally, there are many online resources and forums where you can ask for help with specific physics problems. Keep persevering and seeking understanding, as that is a key aspect of being a successful scientist.